i-MARK: A New Subtraction Division Game
Eric Sopena (LaBRI)

TL;DR
This paper introduces a new subtraction division game called i-MARK, analyzes its Sprague-Grundy sequence, and demonstrates that it exhibits almost periodicity and efficient computability of winning positions.
Contribution
It provides the first analysis of i-MARK games, showing almost periodicity of Sprague-Grundy sequences and efficient computation methods, extending prior work on similar subtraction division games.
Findings
Sprague-Grundy sequence is almost periodic in many cases.
Set of winning positions is periodic.
Winning positions can be computed in O(log n) time.
Abstract
Given two finite sets of integers and ,the impartial combinatorial game is played on a heap of tokens. From a heap of tokens, each player can moveeither to a heap of tokens for some , or to a heap of tokensfor some if divides .Such games can be considered as an integral variant of \MARK-type games, introduced by Elwyn Berlekamp and Joe Buhlerand studied by Aviezri Fraenkel and Alan Guo, for which it is allowed to move from a heap of tokensto a heap of tokens for any .Under normal convention, it is observed that the Sprague-Grundy sequence of the game is aperiodic for any sets and .However, we prove that, in many cases, this sequence is almost periodic and that the set of winning positions is periodic.Moreover, in all…
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Taxonomy
TopicsArtificial Intelligence in Games
